How to Measure Disorder That Never Settles

Original: "Perspective: Measuring physical entropy out of equilibrium"

arXiv:2604.11953 · 2026-04-13 · CC BY 4.0 · 2 min

Scientists have learned to estimate entropy in living systems by compressing data and watching particles.

Tidying a room where children are playing feels endless: you have just put the toys away, and they are scattered again. The same goes for disorder in living systems: bacterial swarms, shifting sand, swarms of fireflies. Usually disorder (entropy) is measured when everything has frozen and settled. But life does not freeze, and the old methods fail here.

To estimate the disorder in such a system, you would have to go through every possible position and velocity of the particles. For a hundred particles that number has a hundred zeros. No computer can handle it. So scientists came up with workarounds. One of them is data compression. If you write all the particle positions to a file and compress it with an archiver, the length of the compressed file shows how much repetition there is in the data. The shorter the file, the less disorder. It is like tidying up: if all the toys are in one box, you can describe the room briefly, "everything is in the box." If they are scattered, the description will be long.

When a bacterial swarm shifts from chaotic motion to coordinated motion, entropy drops sharply, as if someone flipped a switch. What matters here is the directions of motion, not the positions themselves: the bacteria start swimming in one direction without clumping together.

Another route is to watch how particles wander apart. If they spread out quickly, disorder grows. If they stay together, disorder is smaller. These methods have already helped find new transitions in bacterial swarms and in jammed droplets. They went unnoticed before because researchers looked at the overall picture rather than at the details of the disorder.

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The entropy of a continuous distribution can be negative, unlike the discrete case. You can see this with a narrow peak: the narrower the peak, the lower the entropy, and in the limit it goes to minus infinity.

H[P_s] = -\sum_s P_s \ln P_s

A measure of uncertainty or information in a probability distribution over states

h - h_{id} \le \frac{1}{2\rho} \int \frac{d\mathbf{q}}{(2\pi)^d} \mathrm{tr}[\ln C(\mathbf{q}) - I + C(\mathbf{q})]

Maximum entropy given pair correlations, expressed through the correlation matrix in Fourier space

h - h_0 \le \frac{d}{2} \ln \frac{D\tau}{D_0\tau_0}

Upper bound on entropy via the self-diffusion coefficient and relaxation time

Scientists

Claude Shannon Jacob Bekenstein Stephen Hawking Ludwig Boltzmann Rudolf Clausius John Stewart Bell

Concepts

entropy enthalpy non equilibrium state boltzmann entropy non equilibrium Boltzmann equation temperature thermodynamics effective temperature information theory free energy entropic gravity zeroth law of thermodynamics observable equation of state velocity nonlinear dynamics observer energy distribution thermodynamic stability

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How to Measure Disorder That Never Settles